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26.06.2023 / 13:15 - 14:15 / Zeuthen,
Lattice seminarMultigrid Multilevel Monte Carlo and Deflation
Dr. Gustavo Ramirez-Hidalgo
In many applications, the trace of a sparse and large matrix needs to be computed. In particular, in lattice quantum chromodynamics (LQCD), the trace of the inverse of the discretized Dirac operator appears in the disconnected fermion loop contribution to an observable. As simulation methods get more and more precise, these contributions become increasingly important. Hence, we consider here the problem of computing the trace tr(D−1), with D the Dirac operator. The Hutchinson method, which is frequently used to stochastically estimate the trace of the function of a matrix, approximates the trace as the average over estimates of the form xHD−1x. Here, the entries of the vector x follow a certain probability distribution. For N samples, the accuracy for such a Monte Carlo approximation is O(1/N1/2). In recent work, we have introduced multigrid multilevel Monte Carlo: having a multigrid hierarchy with operators Di, Pi and Ri, for level i, we can rewrite the trace in the form tr(D−10)=∑L−1i=0tr(D−1i−PiD−1i+1Ri)+tr(D−1L) (this reduced expression is in the special case when RiPi=I). We can take, in the particular case of LQCD, D0 to be the Dirac operator D. For some cases, for example the standard 2D Laplacian, we have seen significant reductions in the variance and the total computational work with respect to exactly-deflated Hutchinson. In this talk, we explore the use of (exact and inexact) deflation in combination with the multigrid multilevel Monte Carlo method, and demonstrate how this leads to both algorithmic and computational gains in comparison with exactly-deflated Hutchinson. Our implementations and tests make use of High Performance Computing, and the results shown are relevant to LQCD.
More Information: https://indico.desy.de/category/873/
